It\^o perspective on variance renormalisation

We show that the It\^o solutions of the nonlinear stochastic heat equation $$\partial_t u^\varepsilon- \Delta u^\varepsilon =\varepsilon^{3/4} g (u^\varepsilon) \nabla \xi_\varepsilon,$$ where $\xi_\varepsilon$ denotes the mollification in space at scale $\varepsilon>0$ of a space-time white noise $\xi$, converge in law, as $\varepsilon\to 0$, to the solution of the stochastic heat equation with right-hand side $cg'g(u)\xi$ with a constant $c>0$. Since the noise $\nabla\xi$ is supercritical, the small prefactor is not unexpected to obtain a limit, but the exponent $3/4$ is not predicted by naive scaling arguments. The case $g(u)=u$, modulo a Cole-Hopf transform, corresponds to the result of [Hai25] for the KPZ equation. Our argument is relatively short and relies solely on stochastic analytic techniques.